On the Complexity of Hazard-free Circuits

The problem of constructing hazard-free Boolean circuits dates back to the 1940s and is an important problem in circuit design. Our main lower-bound result unconditionally shows the existence of functions whose circuit complexity is polynomially bounded while every hazard-free implementation is prov...

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Bibliographic Details
Published in:Journal of the ACM 2019-08, Vol.66 (4), p.1-20
Main Authors: Ikenmeyer, Christian, Komarath, Balagopal, Lenzen, Christoph, Lysikov, Vladimir, Mokhov, Andrey, Sreenivasaiah, Karteek
Format: Article
Language:English
Subjects:
Algorithms
Boolean
Boolean algebra
Boolean functions
Circuit design
Circuits
Complexity
Gates (circuits)
Hazards
Lower bounds
Monotone functions
Multiplication
Upper bounds
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Summary:The problem of constructing hazard-free Boolean circuits dates back to the 1940s and is an important problem in circuit design. Our main lower-bound result unconditionally shows the existence of functions whose circuit complexity is polynomially bounded while every hazard-free implementation is provably of exponential size. Previous lower bounds on the hazard-free complexity were only valid for depth 2 circuits. The same proof method yields that every subcubic implementation of Boolean matrix multiplication must have hazards. These results follow from a crucial structural insight: Hazard-free complexity is a natural generalization of monotone complexity to all (not necessarily monotone) Boolean functions. Thus, we can apply known monotone complexity lower bounds to find lower bounds on the hazard-free complexity. We also lift these methods from the monotone setting to prove exponential hazard-free complexity lower bounds for non-monotone functions. As our main upper-bound result, we show how to efficiently convert a Boolean circuit into a bounded-bit hazard-free circuit with only a polynomially large blow-up in the number of gates. Previously, the best known method yielded exponentially large circuits in the worst case, so our algorithm gives an exponential improvement. As a side result, we establish the NP-completeness of several hazard detection problems.
ISSN:0004-5411
1557-735X
DOI:10.1145/3320123